# Hooke's Law should be exponential.

• JCienfuegos
In summary, Hooke's law states that strain is proportional to stress, but it may not be uniform in a large spring due to varying tension. However, Hooke's law can still be applied to find the deformation of an elastic bar or cable under its own weight, unless the deformation is too large and non-elastic behavior occurs.
JCienfuegos
I was just thinking, shouldn't hooke's law be represented with an exponential equation?
I am thinking that in class, the springs we use for the hooke's law experiment are small, but if they were very large, then the higher coils would be sustaining much more weight than the lower coils, and would thus be stretched more. I think this would lend itself more to an exponential model.

I guess the ideal model spring represented in Mr Hooke's law is one that is perfectly elastic and quite weightless.

Feel free to come up with a more accurate model for the ones in your lab. But be warned, you'll then have to look for some other plausible reason for your measurements not tallying so well with theory. Inaccurate modelling is always such a convenient excuse for general disagreement between theory and practice!

If you are familiar with Taylor series expansions, then you can understand that any system with a non-vanishing linear term in the expansion will behave close to linear for a perturbation from the equilibrium point sufficiently small. So, Hooke's law is trivial, but it works pretty well in practice.

JCienfuegos said:
I was just thinking, shouldn't hooke's law be represented with an exponential equation?
I am thinking that in class, the springs we use for the hooke's law experiment are small, but if they were very large, then the higher coils would be sustaining much more weight than the lower coils, and would thus be stretched more. I think this would lend itself more to an exponential model.
Hooke's law states that the strain (deformation) is proportional to the stress (tension, force). How will this be modified by the weight of the spring?
The deformation won't be uniform (as it is for weightless or horizontal spring on a table) but Hooke's law still applies as it is. In each piece of spring, the deformation will be proportional to the tension in that region. However the tension varies along the spring.
You can apply Hooke's law to find the deformation of an elastic bar or cable under its own weight.

Only if the deformation is too large Hooke's law will fail but this is not really related to the size of the spring or the existence of gravity. If you take a very small spring and you pull it hard enough, it won't get back to original size and shape. This is an example of non-elastic behavior.

I appreciate your critical thinking and questioning of established theories. However, I must clarify that Hooke's Law is based on the principle of proportionality, which states that the force applied to a spring is directly proportional to the amount of stretch or compression it experiences. This is represented by a linear equation, F = kx, where F is the force, k is the spring constant, and x is the displacement.

While it is true that in larger springs, the higher coils may experience more weight and therefore be stretched more, this does not necessarily mean that Hooke's Law should be represented with an exponential equation. In fact, as long as the principle of proportionality holds true, the relationship between force and displacement will remain linear.

Furthermore, the use of smaller springs in the classroom setting is simply for practicality and ease of experimentation. Hooke's Law has been proven to hold true for a wide range of spring sizes and materials, and the linear equation has been extensively tested and validated through experiments.

In conclusion, Hooke's Law should remain represented with a linear equation, as it accurately describes the relationship between force and displacement in springs of all sizes. I encourage you to continue questioning and exploring scientific concepts, but always base your conclusions on solid evidence and scientific principles.

## 1. What is Hooke's Law?

Hooke's Law is a physical principle that describes the relationship between the force applied to a spring and the resulting displacement of the spring. It states that the force applied is directly proportional to the displacement of the spring, as long as the spring is not stretched or compressed beyond its elastic limit.

## 2. Why should Hooke's Law be exponential?

Hooke's Law should be exponential because it is based on the assumption that the spring is linearly elastic, meaning that the force and displacement are directly proportional. However, in reality, most springs exhibit non-linear behavior when stretched or compressed beyond their elastic limit. This non-linear behavior can be better described by an exponential function.

## 3. How is Hooke's Law related to the concept of elasticity?

Hooke's Law is directly related to the concept of elasticity, as it describes the elastic behavior of a material. Elasticity is the ability of a material to return to its original shape and size after being deformed by an external force. Hooke's Law is a fundamental principle used to study and understand the elastic behavior of materials.

## 4. What are some real-life applications of Hooke's Law?

Hooke's Law has many real-life applications, such as in the design of springs for various mechanical systems, including car suspensions, door hinges, and mattress coils. It is also used in the design of elastic materials, such as rubber bands, bungee cords, and trampolines.

## 5. Are there any limitations to Hooke's Law?

Yes, there are some limitations to Hooke's Law. It only applies to linearly elastic materials, meaning that the force and displacement must be directly proportional. It also does not take into account other factors that may affect the behavior of a spring, such as temperature and material fatigue. Additionally, Hooke's Law may not accurately describe the behavior of materials when they are stretched or compressed beyond their elastic limit.

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